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Affine Primitive Groups and Semisymmetric Graphs

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Affine Primitive Groups and Semisymmetric Graphs Affine primitive groups and Semisymmetric graphs Hua Han Zai Ping Lu∗ Center for Combinatorics Center for Combinatorics LPMC, Nankai University LPMC, Nankai University Tianjin 300071, P. R. China Tianjin 300071, P. R. China [email protected] [email protected] Submitted: Jul 14, 2012; Accepted: May 21, 2013; Published: May 31, 2013 Mathematics Subject Classifications: 05C25, 20B25 Abstract In this paper, we investigate semisymmetric graphs which arise from affine prim- itive permutation groups. We give a characterization of such graphs, and then con- struct an infinite family of semisymmetric graphs which contains the Gray graph as the third smallest member. Then, as a consequence, we obtain a factorization of the complete bipartite graph Kpspt ;pspt into connected semisymmetric graphs, where p is an prime, 1 6 t 6 s with s > 2 while p = 2. Keywords: semisymmetric graph; normal quotient; primitive permutation group 1 Introduction All graphs considered in this paper are assumed to be finite and simple with non-empty edge sets. For a graph Γ , denote by V Γ , EΓ and AutΓ the vertex set, edge set and automorphism group, respectively. A graph Γ is said to be vertex-transitive or edge-transitive if AutΓ acts transitively on V Γ or EΓ , respectively. A regular graph is called semisymmetric if it is edge-transitive but not vertex-transitive. For a graph Γ , an arc of Γ is an ordered pair (α; β) of two adjacency vertices. A graph Γ is called symmetric if it has no isolated vertices and AutΓ acts transitively on the set of arcs of Γ . The class of semisymmetric graphs was first studied by Folkman [9], who possed several open problem. Afterwards, many authors have done much work on this topic, see [1, 2, 3, 7, 8, 10, 11, 12, 15, 16, 17, 18, 19] for references. In particular, lots of interesting examples of such graphs were found. For example, the Folkman graph on 20 vertices, the smallest semisymmetric graph, was constructed by Folkman [9]; the Gray graph, a ∗Supported in part by the NSF of China. the electronic journal of combinatorics 20(2) (2013), #P39 1 cubic graph of order 54, was first observed to be semisymmetric by Bouwer [1]. In 1985, Iofinova and Ivanov [11] classified all bi-primitive cubic semisymmetric graphs and they proved that there are only five such graphs. In this paper, we consider the semisymmetric graphs whose automorphism group con- tains a subgroup inducing an affine primitive permutation group. For more information about groups of this kind, see [5, 6]. To state our result, we need to introduce several concepts and some notation. Let p be a prime and Fp the field of order p. Then, for an integer l > 1 and an irreducible subgroup H of the general linear group GL(l; p), all affine transformations τh;v l form a primitive subgroup X of AGL(l; p) (acting on the vectors of Fp), where h 2 H, l l l h v 2 Fp and τh;v is defined as Fp ! Fp; u 7! u + v. The above group is a split extension l of a regular normal subgroup fτ1;v j v 2 Fpg by the subgroup H. For convenience, for a l subspace V of Fp, we set T(V ) = fτ1;v j v 2 V g and denote by NH (V ) the subgroup of H l fixing V set-wise. Then X = T(Fp):H, and NH (V ) = NH (T(V )), the normalizer of T(V ) in H. For a graph Γ and a subgroup G 6 AutΓ , Γ is said to be G-vertex-transitive or G-edge-transitive if G is transitive on V Γ or EΓ , respectively. The graph Γ is called G-semisymmetric graphs if it is regular, G-edge-transitive but not G-vertex-transitive. Let Γ be a connected G-edge-transitive graph, where G 6 AutΓ . Assume that G is not transitive on V Γ . Then Γ is a bipartite graph with two parts, say U and W , which are the G-orbits on V Γ . Denote respectively by GU and GW the permutation groups induced by G on U and on W . Our main result deals with the case where one of GU and GW is an affine primitive permutation group. l Theorem 1.1. Let X = T(Fp):H, where p is a prime, l > 1 and H is an irreducible subgroup of the general linear group GL(l; p). Then the following two statements are equivalent. l (1) Fp has an s-dimensional subspace V for some integer 1 6 s < l such that jH : t NH (V )j = p for some integer 1 6 t 6 s. (2) There exists a semisymmetric graph Γ with bipartition V Γ = U [ W with one of GU and GW is permutation isomorphic to X for some edge-transitive subgroup G of AutΓ. Remark. Theorem 1.1 suggests the following interesting problem. Problem 1. Characterize irreducible subgroups of the general linear group GL(l; p) satisfying Theorem 1.1 (1). It is well-known that there are no semisymmetric graph of orders 16, 2p and 2p2, see [9]. Thus, for Theorem 1.1 (1), we have l > 3 and (p; l) 6= (2; 3). 2 Proof of Theorem 1.1 Assume that Γ is a G-edge-transitive but not G-vertex-transitive graph, where G 6 AutΓ . Then Γ is a bipartite graph with two parts, say U and W , which are the G-orbits on V Γ . the electronic journal of combinatorics 20(2) (2013), #P39 2 It follows that Γ is semiregular, that is, the vertices in a same bipartition subset have the same valency. For a given vertex α 2 V Γ , the stabilizer acts transitively on Γ (α). Thus, if Γ is vertex-transitive then it must be symmetric. Take β 2 Γ (α). Then each vertex of Γ can be written as αg or βg for some g 2 G. Then, for two arbitrary vertices αg and βh, they hg−1 −1 are adjacent in Γ if and only if α and β are adjacent, i.e., hg 2 GβGα. Moreover, it is well-known and easily shown that Γ is connected if and only if hGα;Gβi = G. Let Γ be a G-semisymmetric graph with two bipartition subsets U and W . Suppose that G has a subgroup R which is regular on both U and W . Take an edge fα; βg 2 EΓ . Then each vertex in U (W , resp.) can be written uniquely as αx (βx, resp.) for some x 2 R. Set S = fs 2 R j βs 2 Γ (α)g. Then αx and βy are adjacent if and only if yx−1 2 S. If R is abelian, then it is easily shown that αx 7! βx−1 ; βx 7! αx−1 ; 8 x 2 R is an automorphism of Γ , which leads to the vertex-transitivity of Γ , refer to [8, 14]. Lemma 2.1. Let Γ be a G-semisymmetric graph. Assume that G has an abelian subgroup which is regular on both parts of Γ . Then Γ is symmetric. Let Γ be a G-semisymmetric graph. Suppose that G has a normal subgroup N which acts intransitively on at least one of the bipartition subsets of Γ . Then we define the 0 quotient graph ΓN to have vertices the N-orbits on V Γ , and two N-orbits ∆ and ∆ are 0 adjacent in ΓN if and only if some α 2 ∆ and some β 2 ∆ are adjacent in Γ . It is easy to see that the quotient ΓN is a regular graph if and only if all N-orbits have the same length. Let Γ be a finite connected G-semisymmetric graph with G 6 AutΓ . Take an edge fα; βg 2 EΓ and let U = αG and W = βG be the two G-orbits on V Γ . Assume that G is unfaithful on U, and let K be the kernel of G acting on U. Then K is faithful on W , and each K-orbits on W contains at least two vertices. It follows that there are two distinct vertices in W which have the same neighborhood in Γ . Thus, as observed in [8], if any two distinct vertices in U have different neighborhoods in the quotient ΓK then Γ is semisymmetric and can be reconstructed from ΓK as follows. Construction 2.2. Let Σ be a bipartite graph with two bipartition subset U and W¯ such that mjW¯ j = jUj for integer m > 1. Define a bipartite graph Σ1;m with vertex ¯ set U [ (W ×Zm) such that α and (β; i) are adjacent if and only if fα; βg 2 EΓ . For convenience, we set Σ1;1 = Σ. For a group X, the socle of X, denoted by soc(X), is generated by all minimal normal subgroups of X. A permutation group is called quasiprimitive if each of its minimal normal subgroups is transitive. Lemma 2.3. Let Γ be a finite connected G-semisymmetric graph with G 6 AutΓ . Take an edge fα; βg 2 EΓ and set U = αG and W = βG. Assume that GU is quasiprimitive. Then either G is faithful on both U and W , or one of the following statements hold. (1) Γ is isomorphic to the complete bipartite graph KjUj;jUj; the electronic journal of combinatorics 20(2) (2013), #P39 3 (2) G is faithful on W but not faithful on U, GU ∼= GW¯ , and Γ is semisymmetric if further GU is primitive, where K is the kernel of G on U and W¯ is the set of K-orbits on W .
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